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## Tentative Lecture Plan

• From the outset this is the plan of 2017.
• There will be some changes. I will update the plan as I lecture.
Week Topic Chapter Pages Remarks
2 Introducution
The basics:
Metric and Banach spaces, duals, $\ell^p$,
Hahn-Banach and consequences
1
2

p. 5-8,
Theorem A.1 (appendix),
Brezis chp 1.1: Corollay 1.2-1.4

Introduction meeting.
1 lecture.
3 Conclusion to Hahn-Banach,
separable and reflexive spaces,

Compactness in metric spaces,
weak convergence, relation to strong convergence,
Banach-Steinhaus without proof

p. 7-10, 15-16
4 Weak compactnes, Eberlein-Smuljan's thm,
weak * convergence, Helley's thm,

Alaoglu's thm, strong compactness for functions: Arzela-Ascoli's thm

p 11-17

5 Distribution theory
Definitions, properties,
operations, regular/singular

operations(cont.), derivatives of regular distr.,
the fundamental theorem, convolution

3

Remark: We define convolution both as a function as in Holden and as a distribution (see e.g. Wikipedia). The two definitions are related, one is the "density function" of the other.
6 Convolutions (cont.), convergence,
approximations

Primitive in 1D, equations in $D'$,
fundamental solutions
7 Lebesgue Spaces
Strong and weak $L^p$, properties, inequalities.

Convolutions, approximation in $L^p$.

Approximation in $L^p$, compactness in $L^p$
4 4.1, Prop 4.4 and Thm 4.6,
4.2, 4.7, 4.3

p 86-89
Proof of Kolmogorov: We take the classical proof and not the one in the Holden notes, see e.g. Theorem A.5 in Holden-Risebro: Front Tracking for Hyperbolic Conservation Laws.

The classical proof is an approximation argument that reduces the proof to an application of Arzela-Ascoli.

OBS: 3 lectures this week, only one next week.

8 Modes of convergence p 59-63 Only one lecture this week, three last week.

9 Convergence and compactness in $L^p,\ p\in(1,\infty)$.

Convergence and compactness in$L^1$,
Dunford-Pettis with proof, uniform integrability, de la Vallee-Poussin.

Convergence and compactness in $L^\infty$
Examples.
p 60-65

p 65-75

My lecture notes on Dunford-Pettis and equiintegrability PDF

Note that the proof of Dunford-Pettis in the notes of Holden lack the conclusion.

The discussion on equiintegrability can not be found in Holden, see my notes.
10 Radon measures, the space $\mathcal M$,
Weak * compactness in $\mathcal M$ and the subspace $L^1$

Sobolev Spaces
Definitions, smooth approximations

5
p 75-79

95-97

This material is partly taken from Folland: Real Analysis chp 7, partly from Holden.

Note: Riesz representation theorem, the space $M$ needs be defined as the space of finite Radon measures.

My lecture notes on Radon measure and compactness: PDF.

11 Smooth approximations (cont.)
straightening the boundary, extensions

Extensions, restrictions/trace
97-97, 179 (App B.3)

97-99
OBS: Global approx up to boundary - the proof in Holden using straightening is not optimal and only gives $C^1$ approximate functions. In the lectures I used the proof from Evans: PDEs chapter 5 that avoids straightening and give $C^\infty$ approximations.

My lecture notes on smooth approximation up the boundary and straightening the boundary: PDF.

12 Restrictions/trace (cont),
Sobolev inequalities, Gagliardo-Nirenberg-Sobolev

Sobolev inequalities: Gagliardo, Poincare.

H\"older spaces, Morrey's inequality.

98-102

102-104

105-107, 111-112

110-111, 118.
Obs: 3 lectures this week.
13 General Sobolev inequalities,
embedding, compactness in $W^{1,p}$

116-118, 18-19, 112-114

Obs: 1 only one lecture this week.

14 Compactness (cont.),
Sobolev chain rule, finite differences.

114-116, 126, 128

Rellich-Kondrachov: In the lectures I give a stronger version of this result than presented in the notes of Holden. I follow the book of Evans, and show compact embedding into Hoelder spaces $C^{0\,\gamma}$.

The proof of the first part of Rellich-Kondarchov's compactness theorem follows from extension, Kolmogorov-Riesz compactness theorem, and interpolation in $L^p$. This way avoids the long regularization + Arzela-Ascoli argument used in the Holden notes (and in PDE book by Evans). In fact the the regularization + Arzela-Ascoli argument is exactly the argument we used in class to proof Kolmogorov-Riesz in the first place (but the proof in the notes of Holden is slightly different).

My lecture notes from this week:

General Sobolev inequalities and Strong comactness in $W^{1,p}$

Strong comactness, chain rule, and difference quotients

15 Application: To be decided